# You're reading: Travels in a Mathematical World

### Reading aloud and enlarging mathematics

My collaborators at The Aperiodical, Katie Steckles and Christian Perfect, have launched a new mathematics magazine podcast called All Squared. In the first episode, number one1, Katie and Christian speak to Edmund Harriss about speaking mathematics out loud.

Towards the end of the conversation, they speak a little about some of the ambiguity in spoken mathematics and how this might affect blind mathematicians. Particularly, they speak about timing and ambiguity, and Christian gives the example (where the comma is a pause): ‘x plus, y squared’ and ‘x plus y, squared’. The placement of the pause changes the meaning of the formula substantially.

A few minutes ago the conference ‘Supporting Disabled Students in STEM‘ began at the Royal Society. This is the spring conference of the STEM Disability Committee (an alphabet soup collaboration between IOP, RAEng, RSC, SoB, CaSE and RS). I am not attending this, though I was asked a couple of weeks ago for two examples for a session at this.

For the first example, I was asked for a formula that would be a bit difficult to read aloud. The aim is not for an unrealistic expression or to trick the audience, but just for something that might cause ambiguity and mean that the reader would stop and think. Here is the equation (apologies for the use of images throughout):

The exercise, later today, will use two volunteers. One reads the formula aloud and the other, with back turned to the screen, will write it down. Try it now with the person on the next desk or in the next office. I’ll wait.

This is the Taylor expansion of ln(x) around x=2. It is an example from a problem sheet in my first year methods course a few weeks ago, so it isn’t unrealistic. I chose it because there are two particular points of possible ambiguity. The first is in “minus one to the power i plus one”. The second is to have the top be “all over n times two to the n”. Is the “one” included in the power? Is the “two to the n” included in the bottom half of the fraction? There are also some conventions for non-mathematicians in the room to take note of, such as the pronunciation of “ln” and the way we read out “sum over i from one to infinity”. All of this means that to be read well requires a thoughtful, mathematically-trained reader.

If the person reading the equation in the session today does it badly then the other volunteer will write the equation wrong and the point will be made. If they do it well, it will be interesting to hear them speak to the group about what choices they made when reading the equation to reduce the ambiguity.

The second example was a piece of mathematics that would need to be broken over two lines when enlarged, and that would be difficult to do so. Here the grey is supposed to indicate the edge of the page. Have a think about it: how would you break this equation to fit on the page (you can’t make it smaller!)?

Again this is an example from class, a partial fractions question. A naive solution might be to simply cut the equation at the page edge. You can hopefully see in the following image that this is unacceptable, in particular because the 2 on the next line looks like -2.

A more intelligent cut might be to take whole terms onto the next line. However, this may cause confusion because it looks like the (x+3) is part of a separate fraction. Is it one fraction plus another? One fraction multiplied by another? This perhaps isn’t unusable, provided you explain to the reader what you have done, but you are presenting the reader (your student?) with non-standard and, strictly speaking, incorrect mathematical notation. If part of being a mathematician is learning to speak and write mathematics so that other mathematicians can understand you, you are doing damage here.

Here I try to keep the two halves of the fraction together by including in-line cuts. This is still weird, particularly the bottom half, but I could imagine writing something like this by hand (perhaps with a multiplication symbol on the bottom cut) so maybe it’s okay.

Finally, here is what I think I would do. The result is mathematically correct and using fairly standard notation (we wouldn’t ordinarily use multiplication symbols in g(x) but this is unusual rather than strictly wrong).

This is hopefully a good solution, but there are still two problems. Interpreting the two functions back into the partial fractions form is an extra cognitive load for this student (not related to the intended learning outcomes of the question), and producing this formatting is a giant amount of extra work compared with the original code. Here is the original LaTeX:

Find the following integral by first resolving the integrand into its partial fractions.    $$\int \frac{x^3+x^2+x+2}{(x+1)(x-2)(x+3)} \, dx$$

And here is the adapted LaTeX code that produced the last version.

Find the following integral by first resolving the integrand into its partial fractions.    $$\int \frac{f(x)}{g(x)} \, dx \text{,}$$    where    \begin{align*}      f(x)= \, &x^3+x^2\\      &+x+2    \end{align*}    and    \begin{align*}      &g(x)=(x+1)\times\\      &(x-2)\times(x+3)    \end{align*}

The person who adapts this page must be properly trained in mathematics, so they don’t introduce errors into the notation and adapt it sensibly, and in this case they must know LaTeX to be able to write the code. This is an unusual set of skills for a generalist disability support professional, and the maths department might be unable to commit personnel to do this. Sometimes the problems are that mathematics notation is difficult to adapt, and sometimes they are to do with the practicalities of who is able to do the work.

I will leave you with this quote from Christian in the podcast.

It is very rare for someone who is blind to get through to becoming a research mathematician, isn’t it? … So is that because the culture isn’t accommodating or because maths is a thing that really is much easier to understand visually?

Something for you to think about. Listen to the podcast to hear what Edmund thinks.

If you are interested in these issues, Emma Cliffe is running a workshop ‘Mathematical Study Without Pen and Paper: Experiences, Impacts and Options‘ for the Higher Education Academy on 20th March 2013 at Manchester Metropolitan University. Attendance is highly recommended.

1. It is an open question, as far as I know, whether the second episode will be number two or number four.

### "Developing a Healthy Scepticism About Technology in Mathematics Teaching"

I have an article in the current issue of the Journal of Humanistic Mathematics (Vol 3, Issue 1). The title is Developing a Healthy Scepticism About Technology in Mathematics Teaching. This will be a chapter of my PhD thesis and provides some background context. I am following a model in which teaching draws on a body of theory which is based on scholarship as well as reflective evaluation of previous experience. So as well as a literature survey, I present a reflective account of experiences which have taken place alongside, but outside of, my PhD research that have shaped my thinking.

This journal is an online-only, diamond open-access, peer-reviewed journal with an emphasis on “the aesthetic, cultural, historical, literary, pedagogical, philosophical, psychological, and sociological aspects as we look at mathematics as a human endeavor”. They publish “articles that focus mainly on the doing of mathematics, the teaching of mathematics, and the living of mathematics”. (Quotes from the Journal’s About page.)

My article’s synopsis is:

A reflective account is presented of experiences which took place alongside a research project and caused a change in approach to be more sceptical about implementation of learning technology. A critical evaluation is given of a previous e-assessment research project, undertaken from a position of naive enthusiasm for learning technology. Experiences of teaching classes and designing assessment tasks lead to doubts regarding the extent to which the previous project encouraged deep learning and contributed to graduate skills development. Investigations of the benefits of another technology—in-class response systems—lead to revelations about learning technology: its enthusiastic introduction in isolation cannot be expected to produce educational benefit; instead it must address some pedagogic need and should be evaluated against this. Overall, these experiences contribute to a shift away from a naive enthusiasm to an approach based on careful consideration of educational need before technology implementation.

P.S. Sorry the blog has become rather infrequent and quite education-focused. I am currently splitting my time between teaching and writing my thesis, so I have little time for anything else. My employment contract is only to teach until May and my thesis is due in July.

I write to share and invite discussion of something I presented at a conference at Nottingham Trent University last week.

I have been thinking a lot about assessment methods and their advantages and limitations for a chapter I am writing for my PhD thesis. For example, I could set a paper test and mark it by hand, as indeed I set one last week and will be marking it when I finish this post, and this allows me to give a personal touch and assess students’ written work but one downside is that I can’t return marks to students very quickly. I could return marks immediately if I used automated assessment, but then setting the assessment would be more difficult and I may be limited in the range of what I could assess. And so on.

I have been trying to classify these advantages and their paired limitations. My thinking is that by viewing different assessment methods as balanced sets of advantages and limitations we can justify different approaches in different circumstances and, particularly for my PhD, explore the advantage/limitation space for any untapped opportunities, which I won’t go into now (but ask me).

Here is my current list of potential advantages that assessment could access. These advantages are each something that I think that some assessment method can offer. My question is: what am I missing? I would be pleased to receive your thoughts on this in the comments.

• Immediate feedback. This is linked to learning from mistakes, confidence and motivation. It can also prioritise procedural learning over conceptual understanding.
• Detailed, personalised feedback. Though there is much disagreement in what I have read whether a human, who can respond to individual student work, or a computer, which will tirelessly generate worked examples using the context of the question asked, will in practice provide this.
• Individualised assessment. This is achieved through randomisation of questions and is linked to repeated practice, deterring plagiarism, allowing students to discuss the method of a piece of work without the risk of copying or collusion.
• Assessing across the whole syllabus. For example, computers can’t mark every topic.
• Testing application of technique. Whether students can apply some procedure.
• Assessing deep or conceptual learning. For example, open-ended or project work may require a detailed manual review to mark. This is linked to graduate skills development, etc.
• Easy to write new questions. Assume it is easy for a lecturer to write questions that students can answer (it isn’t, but we’re talking principle here). Difficulty is introduced by having to second guess an automated system, or having to second guess students to program misconceptions.
• Quick to set assessments. Assume that writing a test manually takes time. By quickly, I really mean choosing items from a question bank.
• Quick to mark assessments. Assume that marking by hand is not quick, perhaps unless the assessment is very short and student answer format very prescribed, in which case the assessment is limited. This is perhaps linked to problems of consistency and fairness when using multiple markers.
• Easy to monitor students. Clearly marking individual work from every student by hand will give great insight, but here I refer to the ability to gain a snapshot of how individuals and the cohort are doing as a whole with a concept, perhaps very soon after a lecture that introduced that concept has taken place.
• Perception of anonymity. I’ve read that some students are happier to make their mistakes if only a computer knows. This can reduce stress.
• Testing mathematical writing. Clearly requires hand-written work.
• Testing computer skills. Clearly requires use of a computer.

For example, then it might be possible to offer ‘Easy to write new questions’, ‘Assessing deep or conceptual learning’ and ‘Testing mathematical writing’ through a traditional paper-based, hand-marked assessment, but this would preclude, for example, ‘Immediate feedback’.

Similarly, a multiple-choice question bank might offer ‘Quick to set assessments’ and ‘Quick to mark assessments’ at the expense of ‘Assessing across the whole syllabus’ and ‘Assessing deep or conceptual learning’.

And so on. I have loads of these for different assessment types.

My question really is, is there anything missing from my list that might be delivered by an assessment method?

### Podcasting update

I have a job. This is not the podcasting update, but it does affect it! If you have listened to the latest Math/Maths Podcast you will know that I will be lecturing mathematics from January while trying to finish my PhD thesis, and that we will be putting that podcast on hiatus while I do so. This means no more talking to Samuel Hansen for at least six months.

There is something you can do to fill this mathematical podcasting gap, however. Samuel is trying to raise money through a Kickstarter to allow him upgrade his equipment and improve the quality, to pay for the travel to conduct face-to-face interviews and to make this his full-time job so he can concentrate on a regular release schedule, for his work in maths (math) and science communication over at ACMEScience.com.

At Kickstarter, Samuel says:

ACMEScience.com has spent the last four years trying to do something that very few others have ever attempted, create entertaining, insightful, and interesting content about mathematics and science. Started by Samuel Hansen in the beginning of 2009, ACMEScience has produced a pop-culture joke filled mathematical panel show, Combinations and Permutations, a show that interviews everyone from the CEO of a stats driven dating site to a stand up mathematician to Neil deGrasse Tyson, Strongly Connected Components, a show that tells the stories of the fights that behind DNA, dinosaurs, and the shape of the universe, Science Sparring Society, a video interview show that has featured predatory bacteria and crowdsourced questions, ACMEScience News Now, and a series of hour long journeys into the world of competitive AI checkers computers and stories of the most interesting 20th C mathematician and much more, Relatively Prime.

You may remember that Samuel raised money through a Kickstarter before, for the extremely well-received documentary series Relatively Prime. So you might judge this as evidence that he is capable of delivering this project. However, you may also remember that if he doesn’t raise the whole amount he needs then he gets nothing.

There are various pledge levels, with various rewards. Some of these are aimed at the individual who wants to own a piece of the project. Others are aimed at people who want to sponsor/advertise via the shows and get their message out there. Looking at the level of pledges so far, Samuel could really do with a few companies or individuals who want to get a message out to a mathematics or science audience coming forward and pledging some money. Relatively Prime was very well listened to, and you could get your message to a large, focused, engaged set of listeners.

There is not long to go (only four days at time of writing) and it doesn’t look good. So please pitch in and also tell everyone you know via your own blog/podcast/social networks/etc. so that others will support his effort.

Here is the video in which Samuel makes his case. It’s six minutes so at least watch that! The Kickstarter page is ACMEScience.com by Samuel Hansen. Donating is easy through Amazon payments.

### Martin Gardner celebration week in Nottingham

Last Sunday saw the anniversary of the birth of Martin Gardner, and as a celebration, the Gathering for Gardner people planned a world-wide party ‘G4G Celebration of Mind‘. It happened to be Maths Jam night on Tuesday, so we put the Nottingham Maths Jam on the G4G-COM map. Then on Friday three of us had agreed to take a puzzles stall to the Nottingham STEM Pop Up Shop, so I added this to the map as well.

A Celebration of Mind party is supposed to “celebrate the legacy of Martin Gardner on or around Sunday, October 21, 2012 through the enjoyment of [one or more of] Puzzles, Magic, Recreational Math, Lewis Carroll, Skepticism and Rationality”.

At Maths Jam I printed a bunch of flexagon material from the Flexagon Party page. I also had a plan: having finished two jobs in recent years with piles of business cards outstanding, I brought these to try some business card origami. In fact, we decided to make a business card Menger sponge. So we started folding.

Meanwhile, John Read had come equipped with some colourful designs to make hexahexaflexagons.

 John Read’s first hexahexaflexagon of the evening. Designs from flexagon.net.

 John Read’s second hexahexaflexagon. Designs from flexagon.net.

At the same time, Jon made a trihexagon.

 Trying to make a triflexagon

Finally, after much business card folding,we had a Menger sponge* (*not a real one, it being a fractal after all!). Here it is, in Maths Jam-style, balanced on a pint of beer.

 The completed business card Menger sponge

And here’s a shot through the Menger sponge, where a geometry puzzle is being attempted.

 Through the completed business card Menger sponge, some geometry is happening

Here are a couple of the other puzzles that we tried, some from the #MathsJam tag on Twitter:

You have 100 coins, 10 of which are showing heads and 90 of which are showing tails (though these are indistinguishable by touch). Blindfolded, you must divide the coins into an even number of heads and tails.

Which is bigger, 3^(21!) or 2^(31!)?

Then on Friday we made our way to Broadmarsh shopping centre for our afternoon at Nottingham’s STEM Pop Up Shop.

 Nottingham STEM Pop Up Shop welcome notice

Here’s a picture of our stall, with Kathryn Taylor presiding, and in the foreground the posters about Martin Gardner, mathematical games and mathematicians that I had printed. Someone did ask me who Martin was and I explained a little; I think I also convinced him to come to Robin Wilson’s talk on Lewis Carroll next month in Derby.

 Martin Gardner posters on our stall

Here’s the detail of our stall, which we called ‘Solving it like a mathematician’. You can get details of the set of puzzles on my website.

 The STEM Pop Up Shop ‘Solving it like a mathematician’ stall

Looking around the shop, I requisitioned the Alan Turing postcards from the ‘My Favourite Scientist’ set for our stall!

 Alan Turing postcards

Here’s a wider view of the stall, with Kathryn entertaining a customer.

 Kathryn Taylor at our stall

And finally, here’s John Read bewitching a crowd with the loop on a chain trick.

 John Read enthuses a crowd with a ring on a chain

 Thank you for visiting

### Nobel week – a place for mathematics?

In a blog post last week, Alex Bellos said:

It is often said that the reason Alfred Nobel did not endow one of his prizes in mathematics was because his wife was having an affair with a mathematician.
While this story has been debunked it is nevertheless frustrating to mathematicians, especially during Nobel week, that the noblest of the sciences is ignored by the Royal Swedish Academy of Sciences.

As an alternative, Alex offers the Mental Calculation World Cup.

A while ago, I wrote a flippant little piece in which I claimed that “Most of the Nobel Prizes are for Mathematics“. While perfectly valid and interesting mathematics takes place within mathematics itself, it is an interesting aspect of mathematics that its applications take place on the boundaries with, or even within, other disciplines. This creates some issues for those championing mathematics. Some people would like to assess the economic impact of mathematics but this is a difficult task. At what point does an application of mathematics belong to science, engineering or technology?

The point of the IMA Mathematics Matters series of articles, as I understand it, is to show where modern mathematics research has had its impact, even though that impact may be “perfectly hidden in its physical manifestation”. Some people would take the result of a piece of research to be “not mathematics” as soon as it finds an application. Unfortunately, defining a piece of work as ‘not mathematics’ as soon as it is applied is a way to ensure that all measures of economic value of the impact of mathematics are zero; yet it is clearly the case that much of science, engineering and technology would be naught without the mathematics that underpins it.

This is a balance we try to strike when finding stories for the Math/Maths Podcast; many interesting stories, and certainly those more likely to be written up by university press offices or the media, are those which apply mathematics in some other area. How far do we follow a story before declaring it “not mathematics” and turning our attention elsewhere?

It is with this mindset that I view the Nobel Prizes. Much of the work for which the prizes are awarded is underpinned by mathematics. I see ‘Nobel week’ as an opportunity for mathematicians to go in search of the mathematics behind each prize, rather than to retreat and complain about the lack of a prize specifically for mathematics.

### Surds: what are they good for?

Here is a question I was asked:

Why is rearranging equations containing square roots on the curriculum for GCSE? What might it be useful for in later life?

This is a two-part question, one part of which is dynamite. When I put the question to Twitter, Paul Taylor @aPaulTaylor was the first to take the bait:

Is usefulness in later life a necessary condition for inclusion on the GCSE curriculum?

Let’s set that aside for now. Whether usefulness is necessary or not, asking what a topic might be useful for in later life is a perfectly valid question for a fourteen year old who is being asked to study that topic.

Surds is one of those confusing areas that I vaguely remember but have to look at a definition to recall properly. The BBC GCSE Bitesize website has “a square root which cannot be reduced to a whole number” and says “you need to be able to simplify expressions involving surds”. Rearranging surds, then, is the business of noticing that the square root of 12 multiplied by the square root of 3 can be combined to give the square root of 36, which is 6.

Surds, then, are a part of general algebraic fluency. I expected, therefore, that one answer would be that this is the kind of manipulation that helps generally with higher mathematics; though I wonder when such neat numbers arise in reality. I also expected to hear that surds were useful in very efficient computation. I remember once speaking to someone who was programming computers to go on board aeroplanes. These had very limited computing power and needed to work in real time; the programming involved all sorts of mental arithmetic tricks to minimise the complexity of calculations.

For the latter, I am not sure how relevant this is to modern engineering or programming. For the former, it might be that we are including this for every student at GCSE simply as part of the algebraic fluency that we hope of from incoming mathematics students at university. When I put the question to Twitter, two responses reflected my cynicism on this point. When are surds useful in later life?

Other, less cynical responses, were available. Early responses:

• Ian Robinson ‏@IanRobinson said: “it allows you to work with precise fractional values without rounding errors in calculations. Useful for engineering etc.”
Later, Colin Beveridge @icecolbeveridge suggested something similar: “in computing, roots are expensive — if you can consolidate them, you save computing time.”
This rings true for me but it was a mathematically-inclined structural engineer who asked the original question. Is this really used in engineering?
• Christian Perfect @christianp said “anything involving making rectangles” thinking particularly of “carpentry and landscaping“.

I put these suggestions – rounding errors and rectangles – to Twitter.

I think it’s unlikely anyone doing practical work would need the accuracy. Feels more pure Maths than Applied. But is it used? For engineers, landscape, carpentry etc expansion to a few decimal places so you can measure to reasonable accuracy is fine.

Carol Randall ‏@Caro_lann said: “engineering isn’t just measurement! There’s lots of heavy maths involved in getting a B.Eng (and beyond).”

John Read ‏@johndavidread asked: “where in Maths do equations with square roots come up that you’d want to simplify without calculating numerical value?”

To this, Daniel Colquitt ‏@danielcolquitt wrote what on Twitter must be considered an essay, a four tweet message (1, 2, 3, 4):

Very simple examples: Computing the eigenfrequencies of beams, or reciprocal lattice vectors & hence in various Fourier transforms. In this case, exact form is required, decimal expansion will not do. For the beam example, a numerical value can be computed for a given set of parameters, but if you want to know that frequency for *any* set of parameters, you need to know how to hand surds.

On algebraic fluency, Christine Corbett ‏@corbett_inc suggested “the umbrella of ‘simplifying equations'”.

To this, John Read ‏@johndavidread asked: “but then why not teach it as ‘simplifying equations’? No kid had heard of a surd in the 1980’s”.

Daniel Colquitt ‏@danielcolquitt replied: “For GCSE & roots of reals >0, I would tend to agree with you. Complex roots are somewhat different”.

But we’ve swayed back rather close to the dynamite, haven’t we? I’ll stop there.

My sense is that I haven’t had a satisfactory answer really. This sort of rearrangement is good for building up the background knowledge of the undergraduate mathematics student or perhaps engineering student, but no one seems to be claiming they are an engineer who uses this outside of the classroom. No one seems to have claimed this topic develops mathematical thinking in an interesting way, or that engineers who don’t think they are using it really are relying on it in the black box of software, or that the topic somehow contributes to an appreciation of the beauty of mathematics in the teenagers who are learning it. (This may be due to my experiences and the experiences of those who have replied, or the way I have misinterpreted their words.) It may be that there’s a bunch of stuff on the GCSE syllabus just for those who go onto A-level or degree-level mathematics, and perhaps that’s fine, but it would be nice to have a more satisfying answer to give. So, dear reader, are you satisfied with these answers? Do you have a better answer?